Shannon sampling and an inverse problem for the Schrodinger equation on combinatorial graphs
Shannon sampling and an inverse problem for the Schrodinger equation on combinatorial graphs
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组合图上薛定谔方程的香农采样和反演问题
DOI：
10.1109/globalsip.2016.7905857
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发表时间：
2016
期刊：
影响因子：
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通讯作者：
I. Pesenson
中科院分区：
文献类型：
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作者：
I. Pesenson
我们考虑加权组合图G（有限或可数）上的函数，其在时间−∞ < t < ∞上的演化由薛定谔型方程<$g（t，v）/<$t = iΔg（t，v），v <$V（G）控制，右侧是组合拉普拉斯算子。证明了两个Shannon型采样定理，即如果初始数据g（0，v）是带宽≤ ω的Paley-Wiener函数，则对所有t ∈（−∞，∞），v ∈ V（G）可以从集合K × S上的g的值完美地重建，其中K是等间隔的真实的数的足够密集的集合，S是Paley空间的采样集合。带宽≤ ω的维纳函数。我们还考虑了从单个样本g（T，·）重构初始函数g（0，·）的逆问题。
We consider functions on a weighted combinatorial graph G (finite or countable) whose evolution in time −∞ < t < ∞ is governed by the Schrödinger type equation ∂g(t, v)/∂t = iΔg(t, v), v ∊ V (G), with the combinatorial Laplace operator on the right side. Two Shannon-type sampling theorems are proved which imply that if the initial data g(0, v) is a Paley-Wiener function of bandwidth ≤ ω then the solution g(t, v) for all t ∊ (−∞,∞), v ∊ V (G) can be perfectly reconstructed from the values of g on a set K × S where K is a sufficiently dense set of equally spaced real numbers and S is a sampling set for the space of Paley-Wiener function of bandwidth ≤ ω. We also consider an inverse problem of reconstructing the initial function g(0, ·) from a single sample g(T, ·).